A355365 -id:A355365 - cp's OEIS frontend

A355361 G.f. A(x) satisfies: xA(x) = Sum_{n=-oo..+oo} (-1)^n x^(n(n+1)/2) A(x)^n.

1, 1, 5, 26, 136, 746, 4261, 25173, 152596, 943804, 5931561, 37768700, 243124702, 1579577423, 10344340396, 68212177180, 452531832109, 3018280278965, 20227324602249, 136135295125566, 919757424512780, 6235752585125348, 42411283395662960, 289289349007740037

Offset: 0

Paul D. Hanna, Jul 19 2022

nonn

Equals the row sums of triangle A355360: a(n) = Sum_{k=0..n} A355360(n,k) for n >= 0.

			G.f.: A(x) = 1 + x + 5*x^2 + 26*x^3 + 136*x^4 + 746*x^5 + 4261*x^6 + 25173*x^7 + 152596*x^8 + 943804*x^9 + 5931561*x^10 + ...
where
x*A(x) = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x)^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A355360, A355362, A355363, A355364, A355365.

(* Calculation of constants {d,c}: *) {1/r, s*Sqrt[((-1 + s)*(-1 + r*s) * Log[r]*((-1 + s)*(-1 + r*s) * QPolyGamma[0, 1, r] - r*(-1 + s)*(-1 + r*s) * Log[r] *  Derivative[0, 1][QPochhammer][r, r] / QPochhammer[r] + r*Log[r]*QPochhammer[r] * QPochhammer[s, r] * Derivative[0, 1][QPochhammer][1/(r*s), r] + (-1 + r*s)*((1 - s) * QPolyGamma[0, Log[s]/Log[r], r] - Log[r]*(s + r*(-1 + s) * Derivative[0, 1][QPochhammer][s, r] / QPochhammer[s, r])))) / (2* Pi*(-s*(1 + r - 4*r*s + r*(1 + r)*s^2) * Log[r]^2 + (-1 + s)^2 * (-1 + r*s)^2 * QPolyGamma[1, Log[s]/Log[r], r] + (-1 + s)^2 * (-1 + r*s)^2 * QPolyGamma[1, -Log[r*s]/Log[r], r]))]} /. FindRoot[{QPochhammer[r] * QPochhammer[1/(r*s), r] * QPochhammer[s, r] / ((-1 + s)*(-1 + r*s)) == -1, 1/(-1 + s) + 1/(-1 + r*s) + (QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, Log[1/(r*s)]/Log[r], r])/Log[r] == -2}, {r, 1/7}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 19 2024 *)

{a(n) = my(A=[1,1],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*n+9));
A[#A] = polcoeff( x*Ser(A) - sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies:
(1) x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) -x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
(3) x*A(x)*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 7.27539777267340429262199058476266... and c = 0.504162727798216251681995853318... - Vaclav Kotesovec, Jul 20 2022
A(1/d) = 1.78721033673795569... where 1/d = 0.1374495293928845... and d is the value given above by Vaclav Kotesovec. - Paul D. Hanna, Jul 30 2022
Formula (3) can be rewritten as the functional equation QPochhammer(y, x)/(1 - y) * QPochhammer(1/(x*y), x)/(1 - 1/(x*y)) = x*y / QPochhammer(x). - Vaclav Kotesovec, Jan 19 2024

A356500 Coefficients T(n,k) of x^ny^k in A(x,y) for n >= 0, k = 0..3n+1, where A(x,y) satisfies: y = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x,y)^((n-1)^2), as an irregular triangle read by rows.

Original entry on oeis.org

0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 6, 0, 0, 0, 28, 0, 0, 0, 22, 0, 3, 0, 0, 0, 84, 0, 0, 0, 219, 0, 0, 0, 140, 0, 0, 0, 0, 135, 0, 0, 0, 981, 0, 0, 0, 1807, 0, 0, 0, 969, 0, 0, 0, 120, 0, 0, 0, 2568, 0, 0, 0, 10764, 0, 0, 0, 15368, 0, 0, 0, 7084, 0, 0, 54, 0, 0, 0, 4284, 0, 0, 0, 38896, 0, 0, 0, 114240, 0, 0, 0, 133266, 0, 0, 0, 53820, 0, 9, 0, 0, 0, 4662, 0, 0, 0, 94390, 0, 0, 0, 525980, 0, 0, 0, 1187433, 0, 0, 0, 1171390, 0, 0, 0, 420732

Offset: 0

Paul D. Hanna, Aug 09 2022

T(n, 3*n+1) = [x^n*y^(3*n+1)] A(x,y) = binomial(4*n, n)/(3*n + 1) = A002293(n) for n >= 0, where g.f. G(x) of A002293 satisfies: G(x) = 1 + x*G(x)^4.
T(4*n, 1) = A000716(n) for n >= 0 (nonzero terms in column 1).
T(4*n+3, 2) = [x^(4*n+3)*y^2] A(x,y) = 2 * A354655(n+1) for n >= 0, where A354655 equals column 2 of triangle A354650.
T(4*n+2, 3) = [x^(4*n+2)*y^3] A(x,y) = 4 * A354656(n+1) for n >= 0, where A354656 equals column 3 of triangle A354650.
T(2*n, 2*n+1) = A356504(n), for n >= 0.
T(2*n+1, 2*n) = A356505(n) for n >= 0.
T(3*n, n+1) = A356506(n) for n >= 0.
T(3*n+1, n) = A355365(n) where A355365 is the central terms of A355360 = A355360(2*n,n).
Sum_{k=0..3*n+1} T(n, k) = A354248(n) for n >= 0 (row sums).
Sum_{k=0..3*n+1} T(n, k) * 2^k = A356502(n) for n >= 0.
Sum_{k=0..3*n+1} T(n, k) * 3^k = A356503(n) for n >= 0.
Sum_{k=0..3*n+1} T(4*n+1-k, k) = A355872(n+1) for n >= 0 (nonzero antidiagonal sums).
SPECIFIC VALUES.
(V.1) 1 = A(x,y) at x = -exp(-Pi) and y = Pi^(1/4)/gamma(3/4).
(V.2) 1 = A(x,y) at x = -exp(-2*Pi) and y = Pi^(1/4)/gamma(3/4) * (6 + 4*sqrt(2))^(1/4)/2.
(V.3) 1 = A(x,y) at x = -exp(-3*Pi) and y = Pi^(1/4)/gamma(3/4) * (27 + 18*sqrt(3))^(1/4)/3.
(V.4) 1 = A(x,y) at x = -exp(-4*Pi) and y = Pi^(1/4)/gamma(3/4) * (8^(1/4) + 2)/4.
(V.5) 1 = A(x,y) at x = -exp(-sqrt(3)*Pi) and y = gamma(4/3)^(3/2)*3^(13/8)/(Pi*2^(2/3)).

			G.f.: A(x,y) = y + x*(1 + y^4) + x^2*(4*y^3 + 4*y^7) + x^3*(6*y^2 + 28*y^6 + 22*y^10) + x^4*(3*y + 84*y^5 + 219*y^9 + 140*y^13) + x^5*(135*y^4 + 981*y^8 + 1807*y^12 + 969*y^16) + x^6*(120*y^3 + 2568*y^7 + 10764*y^11 + 15368*y^15 + 7084*y^19) + x^7*(54*y^2 + 4284*y^6 + 38896*y^10 + 114240*y^14 + 133266*y^18 + 53820*y^22) + x^8*(9*y + 4662*y^5 + 94390*y^9 + 525980*y^13 + 1187433*y^17 + 1171390*y^21 + 420732*y^25) + x^9*(3250*y^4 + 160965*y^8 + 1670942*y^12 + 6640711*y^16 + 12167001*y^20 + 10399545*y^24 + 3362260*y^28) + ...
such that A = A(x,y) satisfies
y = ... + x^16*A^25 - x^9*A^16 + x^4*A^9 - x*A^4 + A - x + x^4*A - x^9*A^4 + x^16*A^9 - x^25*A^16 +- ... + (-x)^(n^2) * A(x,y)^((n-1)^2) + ...
This irregular triangle of coefficients T(n,k) of x^n*y^k in A(x,y) for n >= 0, k = 0..3*n+1, begins:
  n = 0: [0, 1];
  n = 1: [1, 0, 0, 0, 1];
  n = 2: [0, 0, 0, 4, 0, 0, 0, 4];
  n = 3: [0, 0, 6, 0, 0, 0, 28, 0, 0, 0, 22];
  n = 4: [0, 3, 0, 0, 0, 84, 0, 0, 0, 219, 0, 0, 0, 140];
  n = 5: [0, 0, 0, 0, 135, 0, 0, 0, 981, 0, 0, 0, 1807, 0, 0, 0, 969];
  n = 6: [0, 0, 0, 120, 0, 0, 0, 2568, 0, 0, 0, 10764, 0, 0, 0, 15368, 0, 0, 0, 7084];
  n = 7: [0, 0, 54, 0, 0, 0, 4284, 0, 0, 0, 38896, 0, 0, 0, 114240, 0, 0, 0, 133266, 0, 0, 0, 53820];
  n = 8: [0, 9, 0, 0, 0, 4662, 0, 0, 0, 94390, 0, 0, 0, 525980, 0, 0, 0, 1187433, 0, 0, 0, 1171390, 0, 0, 0, 420732];
  n = 9: [0, 0, 0, 0, 3250, 0, 0, 0, 160965, 0, 0, 0, 1670942, 0, 0, 0, 6640711, 0, 0, 0, 12167001, 0, 0, 0, 10399545, 0, 0, 0, 3362260];
  ...
Reading this triangle by nonzero antidiagonals [x^(4*n+1-k)*y^k] A(x,y) for n >= 0, k = 0..3*n+1, yields triangle A356501:
  [1, 1];
  [0, 3, 6, 4, 1];
  [0, 9, 54, 120, 135, 84, 28, 4];
  [0, 22, 294, 1360, 3250, 4662, 4284, 2568, 981, 219, 22];
  [0, 51, 1260, 10120, 41405, 103020, 170324, 196172, 160965, 94390, 38896, 10764, 1807, 140];
  [0, 108, 4590, 58380, 368145, 1404102, 3587696, 6515712, 8715465, 8763645, 6684744, 3863496, 1670942, 525980, 114240, 15368, 969];
  ...

Paul D. Hanna, Table of n, a(n) for n = 0..3926

Cf. A354248, A356502, A356503, A356504, A356505, A355872.
Cf. A354655, A354656, A355365, A000716, A002293.
Cf. related tables: A356501, A355350, A355360, A355870.

{T(n,k) = my(A=[y],M); for(i=1,n, A = concat(A,0); M = ceil(sqrt(n+1));
A[#A] = -polcoeff( sum(m=-M,M, (-x)^(m^2)*Ser(A)^((m-1)^2)), #A-1)); polcoeff(A[n+1],k,y) }
for(n=0,9, for(k=0,3*n+1, print1(T(n,k),", "));print(""))

G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..3*n+1} T(n,k) * x^n * y^k satisfies:
(1) y = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x,y)^((n+1)^2).
(2) y = A(x,y) * Product_{n>=1} (1 - x^(2*n)*A(x,y)^(2*n)) * (1 - x^(2*n-1)*A(x,y)^(2*n+1)) * (1 - x^(2*n-1)*A(x,y)^(2*n-3)), by the Jacobi triple product identity.
(3) y = (-x) * Product_{n>=1} (1 - x^(2*n)*A(x,y)^(2*n)) * (1 - x^(2*n+1)*A(x,y)^(2*n-1)) * (1 - x^(2*n-3)*A(x,y)^(2*n-1)), by the Jacobi triple product identity.
(4) y = A(x, F(x,y)) where F(x,y) = Sum_{n=-oo..+oo} (-x)^(n^2) * y^((n-1)^2).
(5) 1 = A(x, theta_4(x)) where theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2) is a Jacobi theta function.

A355360 G.f. A(x,y) satisfies: xyA(x,y) = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x,y)^n, with coefficients T(n,k) of x^n*y^k in A(x,y) given as a triangle read by rows.

Original entry on oeis.org

1, 0, 1, 0, 3, 2, 0, 9, 12, 5, 0, 22, 54, 46, 14, 0, 51, 196, 282, 175, 42, 0, 108, 630, 1360, 1365, 666, 132, 0, 221, 1836, 5635, 8190, 6321, 2541, 429, 0, 429, 4984, 20850, 41405, 45326, 28448, 9724, 1430, 0, 810, 12744, 70737, 184527, 270060, 237209, 125532, 37323, 4862, 0, 1479, 31050, 223652, 745745, 1404102, 1625932, 1193116, 546039, 143650, 16796

Offset: 0

Paul D. Hanna, Jul 19 2022

The main diagonal equals A000108, the Catalan numbers.
Conjectures.
(C.1) Column 1 equals A000716, the number of partitions into parts of 3 kinds.
(C.2) Column 2 equals twice A023005, the number of partitions into parts of 6 kinds.
The term T(n,k) is found in row n and column k of this triangle, and can be used to derive the following sequences.
A355361(n) = Sum_{k=0..n} T(n,k) for n >= 0 (row sums).
A355362(n) = Sum_{k=0..n} T(n,k) * 2^k for n >= 0.
A355363(n) = Sum_{k=0..n} T(n,k) * 3^k for n >= 0.
A355364(n) = Sum_{k=0..floor(n/2)} T(n-k,k) for n >= 0 (antidiagonal sums).
A355365(n) = T(2*n,n) for n >= 0 (central terms of this triangle).

			G.f.: A(x,y) = 1 + x*y + x^2*(3*y + 2*y^2) + x^3*(9*y + 12*y^2 + 5*y^3) + x^4*(22*y + 54*y^2 + 46*y^3 + 14*y^4) + x^5*(51*y + 196*y^2 + 282*y^3 + 175*y^4 + 42*y^5) + x^6*(108*y + 630*y^2 + 1360*y^3 + 1365*y^4 + 666*y^5 + 132*y^6) + x^7*(221*y + 1836*y^2 + 5635*y^3 + 8190*y^4 + 6321*y^5 + 2541*y^6 + 429*y^7) + x^8*(429*y + 4984*y^2 + 20850*y^3 + 41405*y^4 + 45326*y^5 + 28448*y^6 + 9724*y^7 + 1430*y^8) + x^9*(810*y + 12744*y^2 + 70737*y^3 + 184527*y^4 + 270060*y^5 + 237209*y^6 + 125532*y^7 + 37323*y^8 + 4862*y^9) + x^10*(1479*y + 31050*y^2 + 223652*y^3 + 745745*y^4 + 1404102*y^5 + 1625932*y^6 + 1193116*y^7 + 546039*y^8 + 143650*y^9 + 16796*y^10) + ...
where
x*y*A(x) = ... - x^10/A(x,y)^5 + x^6/A(x,y)^4 - x^3/A(x,y)^3 + x/A(x,y)^2 - 1/A(x,y) + 1 - x*A(x,y) + x^3*A(x,y)^2 - x^6*A(x,y)^3 + x^10*A(x,y)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x,y)^n + ...
also,
x*y*A(x)*P(x) = (1 - x*A(x,y))*(1 - 1/A(x,y)) * (1 - x^2*A(x,y))*(1 - x/A(x,y)) * (1 - x^3*A(x,y))*(1 - x^2/A(x,y)) * (1 - x^4*A(x,y))*(1 - x^3/A(x,y)) * ... * (1 - x^n*A(x,y))*(1 - x^(n-1)/A(x,y)) * ...
TRIANGLE.
The triangle of coefficients T(n,k) of x^n*y^k in A(x,y), for k = 0..n in row n, begins:
n=0: [1];
n=1: [0, 1];
n=2: [0, 3, 2];
n=3: [0, 9, 12, 5];
n=4: [0, 22, 54, 46, 14];
n=5: [0, 51, 196, 282, 175, 42];
n=6: [0, 108, 630, 1360, 1365, 666, 132];
n=7: [0, 221, 1836, 5635, 8190, 6321, 2541, 429];
n=8: [0, 429, 4984, 20850, 41405, 45326, 28448, 9724, 1430];
n=9: [0, 810, 12744, 70737, 184527, 270060, 237209, 125532, 37323, 4862];
n=10: [0, 1479, 31050, 223652, 745745, 1404102, 1625932, 1193116, 546039, 143650, 16796];
n=11: [0, 2640, 72560, 667005, 2784110, 6565030, 9646462, 9242178, 5826171, 2349490, 554268, 58786];
n=12: [0, 4599, 163632, 1892670, 9729720, 28161819, 51126740, 61555824, 50308245, 27806065, 10023948, 2143428, 208012];
...
in which column 1 appears to equal A000716, the coefficients in P(x)^3,
and column 2 appears to equal twice A023005, the coefficients in P(x)^6,
where P(x) is the partition function and begins
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + ... + A000041(n)*x^n + ...
Also, the power series expansions of P(x)^3 and P(x)^6 begin
P(x)^3 = 1 + 3*x + 9*x^2 + 22*x^3 + 51*x^4 + 108*x^5 + 221*x^6 + 429*x^7 + 810*x^8 + 1479*x^9 + 2640*x^10 + ... + A000716(n)*x^n + ...
P(x)^6 = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 918*x^5 + 2492*x^6 + 6372*x^7 + 15525*x^8 + 36280*x^9 + 81816*x^10 + ... + A023005(n)*x^n + ...
The main diagonal equals the Catalan numbers (A000108), where g.f. C(x) = 1 + x*C(x)^2 begins
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + 4862*x^9 + ... + A000108(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..5150

Cf. A000108 (main diagonal), A000041, A000716, A023005.
Cf. A355361 (y=1), A355362 (y=2), A355363 (y=3), A355364, A355365.
Cf. A355350 (related table).

{T(n,k) = my(A=[1,y],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9));
A[#A] = polcoeff( x*y*Ser(A) - sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));polcoeff(A[n+1],k,y)}
for(n=0,12, for(k=0,n, print1( T(n,k),", "));print(""))

G.f. A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k)*y^k satisfies:
(1) x*y*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.
(2) -x*y*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x,y)^n.
(3) x*y*A(x)*P(x) = Product_{n>=1} (1 - x^n*A(x,y)) * (1 - x^(n-1)/A(x,y)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.
(4) A(x,y) = B(x, y*A(x,y)) and A(x, y/B(x,y)) = B(x,y) where x*y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * B(x,y)^n, and B(x,y) is the g.f. of table A355350.

A355362 G.f. A(x) satisfies: 2xA(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x)^n.

Original entry on oeis.org

1, 2, 14, 106, 852, 7286, 65216, 603714, 5731930, 55506348, 546091942, 5443033448, 54845812094, 557774491672, 5717718435034, 59017814463718, 612873311614338, 6398538141213916, 67121038262747380, 707114126290890810, 7478082640450505012, 79360375914717108922

Offset: 0

Paul D. Hanna, Jul 19 2022

nonn

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 106*x^3 + 852*x^4 + 7286*x^5 + 65216*x^6 + 603714*x^7 + 5731930*x^8 + 55506348*x^9 + 546091942*x^10 + ...
where
2*x*A(x) = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x)^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A355360, A355361, A355363, A355364, A355365.

{a(n) = my(A=[1,2],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*n+9));
A[#A] = polcoeff( 2*x*Ser(A) - sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

a(n) = Sum_{k=0..n} A355360(n,k) * 2^k for n >= 0.
G.f. A(x) satisfies:
(1) 2*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) -2*x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
(3) 2*x*A(x)*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 11.38813717738101179115221618346020026348459... and c = 0.5257715220992591718905720654742321646... - Vaclav Kotesovec, Jul 03 2025

A355363 G.f. A(x) satisfies: 3xA(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x)^n.

Original entry on oeis.org

1, 3, 27, 270, 2928, 33912, 411345, 5159337, 66364326, 870637086, 11604385575, 156697653654, 2139109221960, 29472597414681, 409312118499336, 5723853297702444, 80528723782556475, 1139033786793330429, 16187921479930951917, 231046413762053945958

Offset: 0

Paul D. Hanna, Jul 19 2022

nonn

			G.f.: A(x) = 1 + 3*x + 27*x^2 + 270*x^3 + 2928*x^4 + 33912*x^5 + 411345*x^6 + 5159337*x^7 + 66364326*x^8 + 870637086*x^9 + 11604385575*x^10 + ...
where
3*x*A(x) = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x)^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A355360, A355361, A355362, A355364, A355365.

{a(n) = my(A=[1,3],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*n+9));
A[#A] = polcoeff( 3*x*Ser(A) - sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

a(n) = Sum_{k=0..n} A355360(n,k) * 3^k for n >= 0.
G.f. A(x) satisfies:
(1) 3*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) -3*x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
(3) 3*x*A(x)*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 15.42894386025000237511183711088501557092135179... and c = 0.53592940996364915517082259731565361731654... - Vaclav Kotesovec, Jul 03 2025

A355364 G.f. A(x) satisfies: x^2A(x) = Sum_{n=-oo..+oo} (-1)^n x^(n(n+1)/2) A(x)^n.

Original entry on oeis.org

1, 0, 1, 3, 11, 34, 110, 350, 1147, 3800, 12836, 43929, 152285, 533205, 1883187, 6698612, 23974179, 86258459, 311811314, 1131863444, 4124127216, 15078422405, 55301519095, 203405409935, 750122683729, 2773048061073, 10274442343829, 38147288401915

Offset: 0

Paul D. Hanna, Jul 19 2022

nonn

Equals the antidiagonal sums of A355360; a(n) = Sum_{k=0..n} A355360(n-k,k).

			G.f.: A(x) = 1 + x^2 + 3*x^3 + 11*x^4 + 34*x^5 + 110*x^6 + 350*x^7 + 1147*x^8 + 3800*x^9 + 12836*x^10 + 43929*x^11 + 152285*x^12 + ...
where
x^2*A(x) = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x)^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A355360, A355361, A355362, A355363, A355365.

{a(n) = my(A=[1,0],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*n+9));
A[#A] = polcoeff( x^2*Ser(A) - sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) satisfies:
(1) x^2*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
(2) -x^2*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
(3) x^2*A(x)*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 3.92217771004386918... and c = 0.52890084997249... - Vaclav Kotesovec, Jul 03 2025

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A355361 G.f. A(x) satisfies: x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

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A356500 Coefficients T(n,k) of x^n*y^k in A(x,y) for n >= 0, k = 0..3*n+1, where A(x,y) satisfies: y = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x,y)^((n-1)^2), as an irregular triangle read by rows.

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A355360 G.f. A(x,y) satisfies: x*y*A(x,y) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n, with coefficients T(n,k) of x^n*y^k in A(x,y) given as a triangle read by rows.

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A355362 G.f. A(x) satisfies: 2*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

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A355363 G.f. A(x) satisfies: 3*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

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A355364 G.f. A(x) satisfies: x^2*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

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A355361 G.f. A(x) satisfies: xA(x) = Sum_{n=-oo..+oo} (-1)^n x^(n(n+1)/2) A(x)^n.

A356500 Coefficients T(n,k) of x^ny^k in A(x,y) for n >= 0, k = 0..3n+1, where A(x,y) satisfies: y = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x,y)^((n-1)^2), as an irregular triangle read by rows.

A355360 G.f. A(x,y) satisfies: xyA(x,y) = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x,y)^n, with coefficients T(n,k) of x^n*y^k in A(x,y) given as a triangle read by rows.

A355362 G.f. A(x) satisfies: 2xA(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x)^n.

A355363 G.f. A(x) satisfies: 3xA(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n(n+1)/2) A(x)^n.

A355364 G.f. A(x) satisfies: x^2A(x) = Sum_{n=-oo..+oo} (-1)^n x^(n(n+1)/2) A(x)^n.